Question: A circle with radius $10$ has a sector with a $12^\circ$ central angle. What is the area of the sector? ${100\pi}$ $\color{#9D38BD}{12^\circ}$ ${\dfrac{10}{3}\pi}$ ${10}$
Solution: First, calculate the area of the whole circle. Then the area of the sector is some fraction of the whole circle's area. $A_c = \pi r^2$ $A_c = \pi (10)^2$ $A_c = 100\pi$ The ratio between the sector's central angle $\theta$ and $360^\circ$ is equal to the ratio between the sector's area, $A_s$ , and the whole circle's area, $A_c$ $\dfrac{\theta}{360^\circ} = \dfrac{A_s}{A_c}$ $\dfrac{12^\circ}{360^\circ} = \dfrac{A_s}{100\pi}$ $\dfrac{1}{30} = \dfrac{A_s}{100\pi}$ $\dfrac{1}{30} \times 100\pi = A_s$ $\dfrac{10}{3}\pi = A_s$